In this paper, functional convergence is derived for the partial maxima stochastic processes of multivariate linear processes with weakly dependent heavy-tailed innovations and random coefficients. The convergence takes place in the space of Rd-valued càdlàg functions on [0,1] endowed with the weak Skorokhod M1 topology.
Functional limit theoremmultivariate linear processregular variationextremal processM1 topology60F1760G70This work has been supported by University of Rijeka research grant uniri-iskusni-prirod-23-98.Introduction
Let (Xi)i∈Z be a strictly stationary sequence of random variables, and denote by Mn=max{X1,X2,…,Xn}, n≥1, its partial maxima. The asymptotic distributional behavior of Mn is one of the main objects of interest of classical extreme value theory. When (Xi) is an i.i.d. sequence and there exist constants an>0 and bn such that
P(Mn−bnan≤x)→G(x)asn→∞,
with nondegenerated limit G, the limit belongs to the class of extreme value distributions, see [13]. It is known that generalizations of this result to weak convergence of partial maxima processes in the space of càdlàg functions hold. More precisely, relation (1) implies
an−1(Mn(·)−bn):=an−1(⋁i=1⌊n·⌋Xi−bn)→dY(·)
in the space D([0,1],R) of real-valued càdlàg functions on [0,1] endowed with the Skorokhod J1 topology, with Y being an extremal process generated by G (see [11], and Proposition 4.20 in [13]). Simplifying notation, we sometimes omit brackets and write an−1(Mn−bn)→dY. The convergence in relation (2) also holds for a special class of weakly dependent random variables, the linear or moving averages processes with i.i.d. heavy-tailed innovations and deterministic coefficients (see Proposition 4.28 in [13]).
Recently, it was shown in [9] that the functional convergence in (2) holds for linear processes with i.i.d. heavy-tailed innovations and random coefficients. In this paper we aim to generalize this result in two directions, the first one by studying linear processes with weakly dependent innovations (and random coefficients), and the second one by extending this theory to the multivariate setting. Due to possible clustering of large values, the J1 topology becomes inappropriate, and therefore we will use the weaker Skorokhod M1 topology. This topology works well if all extremes within each cluster of large values have the same sign.
The paper is organized as follows. In Section 2 we introduce basic notions about regular variation, linear processes, point processes and Skorokhod topologies. In Section 3 we derive the weak M1 convergence of the partial maxima stochastic process for finite order multivariate linear processes with weakly dependent heavy-tailed innovations and random coefficients. In Section 4 we extend this result to infinite order multivariate linear processes, and give an example which shows that the convergence in the weak M1 topology in general cannot be replaced by the standard M1 convergence.
Preliminaries
We use superscripts in parentheses to designate vector components and coordinate functions, i.e. a=(a(1),…,a(d))∈Rd and x=(x(1),…,x(d)):[0,1]→Rd. For two vectors a=(a(1),…,a(d)), b=(b(1),…,b(d))∈Rd, a≤b means a(k)≤b(k) for all k=1,…,d. The vector (a(1),…,a(d),b(1),…,b(d)) will be denoted by (a,b), and the vector (a(1),b(1),a(2),b(2),…,a(d),b(d)) will be denoted by (a(i),b(i))i=1,…,d∗. Denote by a∨b the vector (a(1)∨b(1),…,a(d)∨b(d)), where for c,d∈R we put c∨d=max{c,d}. Sometimes for convenience we will denote the vector a by (a(i))i=1,…,d. For a real number c we write ca=(ca(1),…,ca(d)).
Regular variation
The Rd-valued random vector ξ is (multivariate) regularly varying if there exist α>0 and a random vector Θ on the unit sphere Sd−1={x∈Rd:‖x‖=1} in Rd, such that for every u>0,
P(‖ξ‖>ux,ξ/‖ξ‖∈·)P(‖ξ‖>x)→wu−αP(Θ∈·)asx→∞,
where the arrow “→w” denotes the weak convergence of finite measures and ‖·‖ denotes the max-norm on Rd. This definition does not depend on the choice of the norm, since if (3) holds for some norm on Rd, it holds for all norms (of course, with different distributions of Θ). The number α is called the index of regular variation of ξ, and the probability measure P(Θ∈·) is called the spectral measure of ξ with respect to the norm ‖·‖. In the one-dimensional case regular variation is characterized by P(|ξ|>x)=x−αL(x), x>0, for some slowly varying function L and the tail balance condition
limx→∞P(ξ>x)P(|ξ|>x)=p,limx→∞P(ξ<−x)P(|ξ|>x)=q,
where p∈[0,1] and p+q=1.
A strictly stationary Rd-valued random process (ξn)n∈Z is regularly varying with index α>0 if for any nonnegative integer k the kd-dimensional random vector ξ=(ξ1,…,ξk) is multivariate regularly varying with index α. According to [2] the regular variation property of the sequence (ξn) is equivalent to the existence of a process (Yn)n∈Z which satisfies P(‖Y0‖>y)=y−α for y≥1, and
((x−1ξn)n∈Z|‖ξ0‖>x)→fidi(Yn)n∈Zasx→∞,
where “→fidi” denotes convergence of finite-dimensional distributions. The process (Yn) is called the tail process of (ξn).
Linear processes
Let (Zi)i∈Z be a strictly stationary sequence of random vectors in Rd, and assume Z1 is multivariate regularly varying with index α>0. We study multivariate linear processes with random coefficients, defined by
Xi=∑j=0∞CjZi−j,i∈Z,
where (Cj)j≥0 is a sequence of d×d matrices (with real-valued random variables as entries) independent of (Zi) such that the above series is a.s. convergent. One sufficient condition for that is ∑j=0∞E‖Cj‖δ<∞ for some δ<α, 0<δ≤1 (see Section 4.5 in [13]), where for a d×d matrix C=(Ci,j), ‖C‖ denotes the operator norm
‖C‖=sup{‖Cx‖:x∈Rd,‖x‖=1}=maxi=1,…,d∑j=1d|Ci,j|.
Skorokhod topologies
Denote by Dd≡D([0,1],Rd) the space of all right-continuous Rd-valued functions on [0,1] with left limits. For x∈Dd the completed (thick) graph of x is defined as
Gx={(t,z)∈[0,1]×Rd:z∈[[x(t−),x(t)]]},
where x(t−) is the left limit of x at t and [[a,b]] is the product segment, i.e. [[a,b]]=[a(1),b(1)]×⋯×[a(d),b(d)] for a=(a(1),…,a(d)), b=(b(1),…,b(d))∈Rd, and [a(i),b(i)] coincides with the closed interval [a(i)∧b(i),a(i)∨b(i)], with c∧d=min{c,d} for c,d∈R. On the graph Gx we define an order by saying that (t1,z1)≤(t2,z2) if either (i) t1<t2, or (ii) |xj(t1−)−z1(j)|≤|xj(t2−)−z2(j)| for all j=1,2,…,d. A weak parametric representation of the graph Gx is a continuous nondecreasing function (r,u) mapping [0,1] into Gx, with r being the time component and u the spatial component, such that r(0)=0, r(1)=1 and u(1)=x(1). Let Πw(x) denote the set of weak parametric representations of Gx. For x1,x2∈Dd define
dw(x1,x2)=inf{‖r1−r2‖[0,1]∨‖u1−u2‖[0,1]:(ri,ui)∈Πw(xi),i=1,2},
where ‖x‖[0,1]=sup{‖x(t)‖:t∈[0,1]}. Now we say that a sequence (xn)n converges to x in Dd in the weak Skorokhod M1 topology if dw(xn,x)→0 as n→∞. If we replace the graph Gx with the completed (thin) graph
Γx={(t,z)∈[0,1]×Rd:z=λx(t−)+(1−λ)x(t)for someλ∈[0,1]},
and weak parametric representations with strong parametric representations, that is continuous nondecreasing functions (r,u) mapping [0,1] onto Γx, then we obtain the standard (or strong) Skorokhod M1 topology. This topology is induced by the metric
dM1(x1,x2)=inf{‖r1−r2‖[0,1]∨‖u1−u2‖[0,1]:(ri,ui)∈Πs(xi),i=1,2},
where Πs(x) is the set of strong parametric representations of the graph Γx. Since Πs(x)⊆Πw(x) for all x∈Dd, the weak M1 topology is weaker than the standard M1 topology on Dd, but they coincide for d=1. The weak M1 topology coincides with the topology induced by the metric
dp(x1,x2)=max{dM1(x1(j),x2(j)):j=1,…,d}
for xi=(xi(1),…,xi(d))∈Dd and i=1,2. The metric dp induces the product topology on Dd.
By using parametric representations in which only the time component r is nondecreasing instead of (r,u) we obtain Skorokhod’s weak and strong M2 topologies. The metric
dM2(x1,x2)=(supa∈Γx1infb∈Γx2d(a,b))∨(supa∈Γx2infb∈Γx1d(a,b)),
where d(a,b)=max{|a(i)−b(i)|:i=1,…,d+1} for a=(a(1),…,a(d+1)), b=(b(1),…,b(d+1))∈Rd+1, induces the strong M2 topology, which is weaker than the M1 topology. For more details and discussion on the M1 and M2 topologies we refer to Sections 12.3-5 and 12.10-11 in [17]. Since the sample paths of the partial maxima processes in (2) are nondecreasing, we will restrict our attention to the subspace D↑d of functions x in Dd for which the coordinate functions x(i) are nondecreasing for all i=1,…,d.
Point processes
Let (Zi)i∈Z be a strictly stationary sequence of regularly varying Rd-valued random vectors with index α>0. Assume the elements of this sequence are pairwise asymptotically (or extremally) independent in the sense that
limx→∞P(‖Zi‖>x,‖Zj‖>x)P(‖Z1‖>x)=0for alli≠j.
We also assume asymptotical independence of the components of each Zi:
limx→∞P(|Zi(j)|>x||Zi(k)|>x)=0for allj,k∈{1,…,d},j≠k.
Condition (7) implies the sequence (Zi) is regularly varying with index α (Proposition 2.1.8 in [10]) with the tail process as in the i.i.d. case, that is Yi=0 for i≠0, and P(‖Y0‖>y)=y−α for y≥1. Relation (8) implies that Y0 a.s. has no two nonzero components.
Define the time-space point processes
Nn=∑i=1nδ(i/n,Zi/an)for alln∈N,
with (an) being a sequence of positive real numbers such that
nP(‖Z1‖>an)→1asn→∞.
The point process convergence for the sequence (Nn) on the space [0,1]×Ed, where Ed=[−∞,∞]d∖{0}, was obtained in [3] under the following two weak dependence conditions.
(Mixing condition).
There exists a sequence of positive integers(rn)such thatrn→∞andrn/n→0asn→∞and such that for every nonnegative continuous function f on[0,1]×Edwith compact support, denotingkn=⌊n/rn⌋, asn→∞,E[exp{−∑i=1nf(in,Zian)}]−∏k=1knE[exp{−∑i=1rnf(krnn,Zian)}]→0.
(Anticlustering condition).
There exists a sequence of positive integers(rn)such thatrn→∞andrn/n→0asn→∞and such that for everyu>0,limm→∞lim supn→∞P(maxm≤|i|≤rn‖Zi‖>uan|‖Z0‖>uan)=0.
The sequences (rn) in these two conditions are assumed to be the same. It can be shown that Condition 2.1 holds for strongly mixing random sequences (see [5, 7]). Now we show that Condition 2.2 holds under condition (7). Let
xk,n:=∑|i|=1kP(‖Zi‖>uan,‖Z0‖>uan)P(‖Z0‖>uan)
for k,n∈N. For every k∈N condition (7) implies that xk,n→0 as n→∞. An application of the triangular argument (Lemma A.1.3 in [10]) yields that there exists a nondecreasing sequence of positive integers (sn)n such that sn→∞ and xsn,n→0 as n→∞. Denote
rn:=min{sn,⌊n⌋},n∈N.
Then rn→∞ and rn/n→0 as n→∞. For a fixed m∈N and large rn (such that rn≥m) it holds that
P(maxm≤|i|≤rn‖Zi‖>uan|‖Z0‖>uan)≤∑|i|=mrnP(‖Zi‖>uan,‖Z0‖>uan)P(‖Z0‖>uan)≤xsn,n,
and letting n→∞ we obtain
limn→∞P(maxm≤|i|≤rn‖Zi‖>uan|‖Z0‖>uan)=0
for every m∈N. Hence, letting m→∞, we see that Condition 2.2 holds. The latter condition holds also under Leadbetter’s condition D′:
limk→∞lim supn→∞n∑i=1⌊n/k⌋P(‖Z0‖>xan,‖Zi‖>xan)=0for allx>0.
The asymptotical independence condition (7) also holds under condition D′. For more discussion of the above weak dependence conditions, in the context of partial sums, we refer to Section 9.1 in [12] and [16].
In the sequel whenever we assume that Condition 2.1 holds we suppose that the sequence (rn) that appears in this condition is the same as in (10), and this will ensure that Conditions 2.1 and 2.2 are satisfied by the same sequence (rn). Under Condition 2.1 by Theorem 3.1 in [3], as n→∞,
Nn→dN=∑i∑jδ(Ti,Piηij)
in [0,1]×Ed, where
∑i=1∞δ(Ti,Pi) is a Poisson process on [0,1]×(0,∞) with intensity measure Leb×ν, with ν(dx)=θαx−α−1dx and θ=P(supi≤−1‖Yi‖≤1).
(∑j=1∞δηij)i is an i.i.d. sequence of point processes in Ed independent of ∑iδ(Ti,Pi) and with common distribution equal to the distribution of the point process ∑jδY˜j/L(Y˜), where L(Y˜)=supj∈Z‖Y˜j‖ and ∑jδY˜j is distributed as (∑j∈ZδYj|supi≤−1‖Yi‖≤1).
Taking into account the form of the tail process (Yi), it holds that θ=1 and N=∑iδ(Ti,Piηi0) with ‖ηi0‖=1. Hence, denoting Qi=ηi0, the limiting point process in relation (12) reduces to
N=∑iδ(Ti,PiQi).
Since the sequence (Qi) is independent of the Poisson process ∑i=1∞δ(Ti,Pi), an application of Proposition 5.3 in [14] yields that ∑iδ(Ti,Pi,Qi) is a Poisson process on [0,1]×(0,∞)×Ed with intensity measure Leb×ν×F, where F is the common probability distribution of Qi.
For x∈R let x+=|x|1{x>0} and x−=|x|1{x<0}. Define the maximum functional Φ:Mp([0,1]×Ed)→D↑2d2 by
Φ(∑iδ(ti,(xi(1),…,xi(d))))(t)=((⋁ti≤txi(j)+,⋁ti≤txi(j)−)j=1,…,d∗)k=1,…,d
for t∈[0,1] (with the convention ∨∅=0), where the space Mp([0,1]×Ed) of Radon point measures on [0,1]×Ed is equipped with the vague topology (see Chapter 3 in [13]). Note that on the right-hand side in (14) we repeat the 2d coordinates of the vector (⋁ti≤txi(j)+,⋁ti≤txi(j)−)j=1,…,d∗ consecutively d times. Let
Λ={η∈Mp([0,1]×Ed):η({0,1}×Ed)=0andη([0,1]×{(x(1),…,x(d)):|x(i)|=∞for somei})=0}.
Then Proposition 3.1 in [9] and the definition of the metric dp in (6) yield the continuity of the maximum functional Φ on the set Λ in the weak M1 topology.
Finite order linear processes
Let (Zi)i∈Z be a strictly stationary sequence of regularly varying Rd-valued random vectors with index α>0. Fix m∈N, and let
Xi=∑j=0mCjZi−j,i∈Z,
be a finite order linear process, where C0,C1,…,Cm are random d×d matrices independent of (Zi). Define the corresponding partial maxima process by
Mn(t)=an−1⋁i=1⌊nt⌋Xi=(an−1⋁i=1⌊nt⌋Xi(k))k=1,…,d,t≥1n,an−1X1=an−1(X1(1),…,X1(d)),t<1n,
for t∈[0,1], with the normalizing sequence (an) as in (9). For k,j∈{1,…,d} let
D+k,j=⋁i=0mCi;k,j+andD−k,j=⋁i=0mCi;k,j−,
where Ci;k,j is the (k,j)th entry of the matrix Ci, Ci;k,j+=|Ci;k,j|1{Ci;k,j>0} and Ci;k,j−=|Ci;k,j|1{Ci;k,j<0}.
First, we show in the proposition below that a particular process Wn, constructed from the sequence (Zi), converges in D↑d with the weak M1 topology. Later, in the main result of this section, we show that the weak M1 distance between processes Mn and Wn is asymptotically negligible (as n→∞), which will imply the functional convergence of the maxima process Mn. The limiting process will be described in terms of certain extremal processes derived from the point process N=∑iδ(Ti,PiQi) in relation (13). Extremal processes can be derived from Poisson processes in the following way. Let ξ=∑kδ(tk,jk) be a Poisson process on [0,∞)×[0,∞)d with mean measure Leb×μ, where μ is a measure on [0,∞)d satisfying μ({x∈[0,∞)d:‖x‖>δ})<∞ for any δ>0. The extremal process G(·) generated by ξ is defined by G(t)=⋁tk≤tjk for t>0. Then for x∈[0,∞)d, x≠0, and t>0 it holds that P(G(t)≤x)=exp(−tμ([[0,x]]c)) (cf. Section 5.6 in [14]). The measure μ is called the exponent measure.
Let(Xi)be a linear process defined in (15), where(Zi)i∈Zis a strictly stationary sequence of regularly varyingRd-valued random vectors with indexα>0that satisfy (7) and (8), andC0,C1,…,Cmare randomd×dmatrices independent of(Zi). Assume Condition2.1holds. LetWn(t):=(⋁i=1⌊nt⌋⋁j=1dan−1(D+k,jZi(j)+∨D−k,jZi(j)−))k=1,…,d,t∈[0,1],withD+k,jandD−k,jdefined in (17). Then, asn→∞,Wn(·)→dM(·):=(⋁j=1d(D˜+k,jM(j+)(·)∨D˜−k,jM(j−)(·)))k=1,…,dinD↑dwith the weakM1topology, whereM(j+)andM(j−)are extremal processes with exponent measuresνj+andνj−respectively, withνj+(dx)=E(Q1(j)+)ααx−α−1dxandνj−(dx)=E(Q1(j)−)ααx−α−1dxforx>0(j=1,…,d), and((D˜+k,j,D˜−k,j)j=1,…,d∗)k=1,…,dis a2d2-dimensional random vector, independent of(M(j+),M(j−))j=1,…,d, such that((D˜+k,j,D˜−k,j)j=1,…,d∗)k=1,…,d=d((D+k,j,D−k,j)j=1,…,d∗)k=1,…,d.
In Proposition 3.1, as well as in the sequel of this paper, we suppose M(j+) is an extremal process if E(Q1(j)+)α>0, and a zero process if this quantity is equal to zero. Analogously for M(j−).
As noted in Subsection 2.4, condition (7) implies that the sequence (Zi) is regularly varying with index α and that Condition 2.2 holds. This, with Condition 2.1, implies the point process convergence in (12) with the limiting point process N described in (13). Since N is a Poisson process, it almost surely belongs to the set Λ. Therefore, since the maximum functional Φ is continuous on Λ, the continuous mapping theorem (see, for instance, Theorem 3.1 in [14]) applied to the convergence in (12) yields Φ(Nn)→dΦ(N) in D↑2d2 under the weak M1 topology, i.e.
Wn⋆(·):=((an−1⋁i=1⌊n·⌋Zi(j)+,an−1⋁i=1⌊n·⌋Zi(j)−)j=1,…,d∗)k=1,…,d→dW(·):=((⋁Ti≤·PiQi(j)+,⋁Ti≤·PiQi(j)−)j=1,…,d∗)k=1,…,d.
By the same arguments as in the proof of Proposition 3.2 in [9] we obtain that D↑2d2 with the weak M1 topology is a Polish space, and hence by Corollary 5.18 in [4] we can find a random vector ((D˜+k,j,D˜−k,j)j=1,…,d∗)k=1,…,d, independent of W, such that
((D˜+k,j,D˜−k,j)j=1,…,d∗)k=1,…,d=d((D+k,j,D−k,j)j=1,…,d∗)k=1,…,d.
This, relation (19) and the fact that ((D+k,j,D−k,j)j=1,…,d∗)k=1,…,d is independent of Wn⋆, by an application of Theorem 3.29 in [4], imply that
(B,Wn⋆)→d(B˜,W)asn→∞
in D↑4d2 with the product M1 topology, where B=((B+k,j,B−k,j)j=1,…,d∗)k=1,…,d and B˜=((B˜+k,j,B˜−k,j)j=1,…,d∗)k=1,…,d are random elements in D↑2d2 such that B+k,j(t)=D+k,j, B−k,j(t)=D−k,j, B˜+k,j(t)=D˜+k,j and B˜−k,j(t)=D˜+k,j for t∈[0,1].
A multivariate version of Lemma 2.1 in [9] implies that the function g:D↑4d2→D↑2d2 defined by
g(x)=(x(1)x(2d2+1),x(2)x(2d2+2),…,x(2d2)x(4d2))
for x=(x(1),…,x(4d2))∈D↑4d2, is continuous in the weak M1 topology on the set of all functions in D↑4d2 for which the first 2d2 component functions have no discontinuity points, and this yields P[(B˜,W)∈Disc(g)]=0, where Disc(g) denotes the set of discontinuity points of g. A multivariate version of Lemma 2.2 in [9] shows that the function h:D↑2d2→D↑d, defined by
h(x)=(⋁i=12dx(i),⋁i=2d+14dx(i),…,⋁i=2(d−1)d+12d2x(i))
for x=(x(i))i=1,…,2d2∈D↑2d2, is continuous when both spaces D↑2d2 and D↑d are endowed with the weak M1 topology. Therefore, the continuous mapping theorem applied to the convergence in (20) yields (h∘g)(B,Wn⋆)→d(h∘g)(B˜,W) as n→∞, i.e.
(⋁i=1⌊n·⌋⋁j=1dD+k,jZi(j)+∨D−k,jZi(j)−an)k=1,…,d→d(⋁Ti≤·⋁j=1d(D˜+k,jPiQi(j)+∨D˜−k,jPiQi(j)−))k=1,…,d
in D↑d with the weak M1 topology. Note that (h∘g)(B,Wn⋆) is equal to Wn.
To finish the proof, it remains to show that (h∘g)(B˜,W) is equal to the limiting process in relation (18). By an application of Propositions 5.2 and 5.3 in [14] we obtain that for every j=1,…,d the point process ∑iδ(Ti,PiQi(j)+) is a Poisson process with intensity measures Leb×νj+, and hence M(j+)(t):=⋁Ti≤tPiQi(j)+ is an extremal processes with exponent measures νj+ (see Section 4.3 in [13]; and [14], p. 161). Analogously, M(j−)(t):=⋁Ti≤tPiQi(j)− is an extremal processes with exponent measures νj−, and hence
(h∘g)(B˜,W)=(⋁j=1d(D˜+k,jM(j+)∨D˜−k,jM(j−)))k=1,…,d,t∈[0,1].
□
The proof of the next theorem relies on the proof of Theorem 3.3 in [9] where the functional convergence of the partial maxima process is established for univariate linear processes with i.i.d. innovations and random coefficients. We will omit some details of those parts of the proof that remain the same in our case, but we will show how to handle those parts that differ due to the multivariate setting and weak dependence of innovations.
Let(Zi)i∈Zbe a strictly stationary sequence of regularly varyingRd-valued random vectors with indexα>0that satisfy (7) and (8), and letC0,C1,…,Cmbe randomd×dmatrices independent of(Zi). Assume Condition2.1holds. ThenMn→dMasn→∞inD↑dendowed with the weakM1topology.
Let Wn be as defined in Proposition 3.1. If we show that for every δ>0,
limn→∞P[dp(Wn,Mn)>δ]=0,
then from Proposition 3.1 by an application of Slutsky’s theorem (see Theorem 3.4 in [14]) it will follow that Mn→dM in D↑d with the weak M1 topology. Taking into account (6) we need to show
limn→∞P[dM1(Wn(j),Mn(j))>δ]=0,
for every j=1,…,d, but it is enough to prove the last relation only for j=1 (since the proof is analogous for all coordinates j). In fact, it suffices to show
limn→∞P[dM2(Wn(1),Mn(1))>δ]=0,
since for x,y∈D↑1 it holds that dM2(x,y)=dM1∗(x,y), where dM1∗ is a complete metric topologically equivalent to dM1 (see Remark 12.8.1 in [17]; and [9], page 247).
In order to show (21), fix δ>0 and let n∈N be large enough, i.e. n>max{2m,2m/δ}. By the definition of the metric dM2 we have
dM2(Wn(1),Mn(1))=(supv∈ΓWn(1)infz∈ΓMn(1)d(v,z))∨(supv∈ΓMn(1)infz∈ΓWn(1)d(v,z))=:Rn∨Tn.
Hence
P[dM2(Wn(1),Mn(1))>δ]≤P(Rn>δ)+P(Tn>δ).
To estimate the first term on the right-hand side of (22), define
Dn={∃v∈ΓWn(1)such thatd(v,z)>δfor everyz∈ΓMn(1)}.
Note that {Rn>δ}⊆Dn. On the event Dn it holds that d(v,ΓMn(1))>δ. Let v=(tv,xv). Then as in the proof of Theorem 3.3 in [9], for all l=0,1,…,m it holds that
|Wn(1)(i∗n)−Mn(1)(i∗+ln)|≥d(v,ΓMn(1))>δ
with i∗=⌊ntv⌋ or i∗=⌊ntv⌋−1. Note that i∗ is a random index. Let D=⋁k,j=1,…,d(D+k,j∨D−k,j). This implies |Ci:k,j|≤D for all i∈{0,…,m} and k,j∈{1,…,d}. Denote δ∗=δ/[8(m+1)d]. We claim that
Dn⊆Hn,1∪Hn,2∪Hn,3,
where
Hn,1={∃l∈{−m,…,m}∪{n−m+1,…,n}s.t.D‖Zl‖an>δ∗},Hn,2={∃k∈{1,…,n}and∃l∈{k−m,…,k+m}∖{k}such thatD‖Zk‖an>δ∗andD‖Zl‖an>δ∗},Hn,3={∃k∈{1,…,n},∃j0∈{1,…,d}and∃p∈{1,…,d}∖{j0}such thatD|Zk(j0)|an>δ∗andD|Zk(p)|an>δ∗}.
Note that relation (24) will be proven if we show that
Dˆn:=Dn∩(Hn,1∪Hn,2∪Hn,3)c=∅.
Assume the event Dˆn occurs. Then necessarily Wn(1)(i∗/n)>δ∗. Indeed, if Wn(1)(i∗/n)≤δ∗, that is
⋁i=1i∗⋁j=1dan−1(D+1,jZi(j)+∨D−1,jZi(j)−)=Wn(1)(i∗n)≤δ∗,
then for every s∈{m+1,…,i∗} it holds that
Xs(1)an=∑r=0m∑j=1dCr;1,jZs−r(j)an≤∑r=0m∑j=1dD+1,jZs−r(j)+∨D−1,jZs−r(j)−an≤∑r=0m∑j=1dδ8(m+1)d=δ8,
since by the definition of D+1,j and D−1,j we have D+1,jZs−r(j)+≥0, D−1,jZs−r(j)−≥0 and
Cr;1,jZs−r(j)≤D+1,jZs−r(j)+,ifCr;1,j>0andZs−r(j)>0,D−1,jZs−r(j)−,ifCr;1,j<0andZs−r(j)<0,0,ifCr;1,j·Zs−r(j)≤0.
Since the event Hn,1c occurs for every s∈{1,…,m}, we also have
|Xs(1)|an≤∑r=0m∑j=1d|Cr;1,j||Zs−r(j)|an≤∑r=0m∑j=1dD‖Zs−r‖an≤(m+1)dδ∗=δ8.
Combining (25) and (26) we obtain
−δ8≤X1(1)an≤Mn(1)(i∗n)=⋁s=1i∗Xs(1)an≤δ8,
and thus
|Wn(1)(i∗n)−Mn(1)(i∗n)|≤|Wn(1)(i∗n)|+|Mn(1)(i∗n)|≤δ8(m+1)d+δ8≤δ4,
which is in contradiction to (23).
Therefore Wn(1)(i∗/n)>δ∗, and hence there exist k∈{1,…,i∗} and j0∈{1,…,d} such that
Wn(1)(i∗n)=an−1(D+1,j0Zk(j0)+∨D−1,j0Zk(j0)−)>δ∗.
This implies
D‖Zk‖an=Dan⋁j=1d|Zk(j)|≥Dan|Zk(j0)|≥1an(D+1,j0Zk(j0)+∨D−1,j0Zk(j0)−)>δ∗.
From this, since Hn,1c∩Hn,2c∩Hn,3c occurs, it follows that m+1≤k≤n−m,
D‖Zl‖an≤δ∗for alll∈{k−m,…,k+m}∖{k},
and
D|Zk(p)|an≤δ∗for allp∈{1,…,d}∖{j0}.
Similarly as in the proof of Theorem 3.3 in [9] one can show that Mn(1)(i∗/n)=Xj(1)/an for some j∈{1,…,i∗}∖{k,…,k+m}. Now we have four cases:
all random vectors Zj−m,…,Zj are “small”,
exactly one is “large” with exactly one “large” component,
exactly one is “large” with at least two “large” components,
at least two of them are “large”,
where we say Z is “large” if D‖Z‖/an>δ∗, otherwise it is “small”, and similarly the component Z(s) is “large” if D|Z(s)|/an>δ∗.
Following the arguments from [9], adjusted to the multivariate setting, it can be shown that the cases (A1) and (A2) are not possible (see the arXiv preprint [6] for details). The case (A3) is not possible on the event Hn,3c, and the case (A4) is not possible on the event Hn,2c. Since neither of the four cases (A1)–(A4) is possible, we conclude that Dˆn=∅, and hence (24) holds.
The next step is to show that P(Hn,k)→0 as n→∞ for k=1,2,3. By stationarity we have P(Hn,1)≤(3m+1)P(D‖Z1‖>δ∗an), and therefore
limn→∞P(Hn,1)=0.
As for Hn,2 we have
P(Hn,2∩{D≤c})=∑k=1n∑l=k−ml≠kk+mP(D‖Zk‖an>δ∗,D‖Zl‖an>δ∗,D≤c)≤2n∑i=1mP(‖Z0‖an>δ∗c,‖Zi‖an>δ∗c)≤2∑i=1mnP(‖Z0‖an>δ∗c)P(‖Z0‖an>δ∗c,‖Zi‖an>δ∗c)P(‖Z0‖an>δ∗c)
for an arbitrary c>0. Therefore regular variation and the asymptotical independence condition (7) yield limn→∞P(Hn,2∩{D≤c})=0, and this implies
lim supn→∞P(Hn,2)≤lim supn→∞P(Hn,2∩{D>c})≤P(D>c).
Letting c→∞ we conclude
limn→∞P(Hn,2)=0.
By the definition of the set Hn,3 and stationarity it holds that
P(Hn,3∩{D≤c})=∑k=1n∑l,s=1l≠sdP(D|Zk(l)|an>δ∗,D|Zk(s)|an>δ∗,D≤c)≤∑l,s=1l≠sdnP(|Z1(s)|an>δ∗c)P(|Z1(l)|an>δ∗c||Z1(s)|an>δ∗c)≤∑l,s=1l≠sdnP(‖Z1‖an>δ∗c)P(|Z1(l)|an>δ∗c||Z1(s)|an>δ∗c),
and hence regular variation and condition (8) yield
limn→∞P(Hn,3)=0.
Now from relations (24) and (29)–(31) we obtain limn→∞P(Dn)=0, and since {Rn>δ}⊆Dn, we conclude that
limn→∞P(Rn>δ)=0.
Interchanging the roles of Mn(1) and Wn(1) in handling the event Dn, and using the arguments from the proof of Theorem 3.3 in [9], adjusted to the multivariate setting, we can show
limn→∞P(Tn>δ)=0
(for details see the arXiv preprint [6]). Now from (22), (32) and (33) we obtain (21), which means that Mn→dM in D↑d with the weak M1 topology. □
Infinite order linear processes
Let (Zi)i∈Z be a strictly stationary sequence of regularly varying Rd-valued random vectors with index α>0, and (Ci)i≥0 a sequence of random d×d matrices independent of (Zi) such that the series defining the linear process
Xi=∑j=0∞CjZi−j,i∈Z,
is a.s. convergent. For k,j∈{1,…,d} let
D+k,j=max{Ci;k,j+:i≥0}andD−k,j=max{Ci;k,j−:i≥0},
where Ci;k,j is the (k,j)th entry of the matrix Ci. Let Mn be the partial maxima process as defined in (16), and M the limiting process from Proposition 3.1.
To obtain functional convergence of the partial maxima process for infinite order linear processes, we first approximate them by a sequence of finite order linear processes, for which Theorem 3.3 holds, and then show that the error of approximation is negligible in the limit with respect to the weak M1 topology. In this case, besides the conditions from Theorem 3.3 for finite order linear processes, we will need also some moment conditions on the sequence of coefficients.
Let(Zi)i∈Zbe a strictly stationary sequence of regularly varyingRd-valued random vectors with indexα>0that satisfy (7) and (8), and let(Ci)i≥0be a sequence of randomd×dmatrices independent of(Zi). Assume Condition2.1holds and suppose∑j=0∞E(‖Cj‖δ+‖Cj‖γ)<∞,ifα∈(0,1),∑j=0∞E(‖Cj‖δ+‖Cj‖)<∞,ifα=1,∑j=0∞E‖Cj‖<∞,ifα>1,for someδ∈(0,α)andγ∈(α,1). ThenMn→dMasn→∞inD↑dendowed with the weakM1topology.
For m∈N, m≥2, define
Xim=∑j=0m−2CjZi−j+C(m,∨)Zi−m+1+C(m,∧)Zi−m,i∈Z,
and
Mn,m(t)=an−1⋁i=1⌊nt⌋Xim,t∈[1n,1],an−1X1m,t∈[0,1n),
where C(m,∨)=max{Ci:i≥m−1} and C(m,∧)=min{Ci:i≥m−1}, with the maximum and minimum of matrices interpreted componentwise, i.e. the (k,j)th entry of the matrix C(m,∨) is Ck,j(m,∨)=max{Ci;k,j:i≥m−1}, and the (k,j)th entry of the matrix C(m,∧) is Ck,j(m,∧)=min{Ci;k,j:i≥m−1}.
For k,j∈{1,…,d} define
D+m,k,j=(⋁i=0m−2Ci;k,j+)∨Ck,j(m,∨)+∨Ck,j(m,∧)+
and
D−m,k,j=(⋁i=0m−2Ci;k,j−)∨Ck,j(m,∨)−∨Ck,j(m,∧)−.
Then D+m,k,j=D+k,j and D−m,k,j=D−k,j, and therefore for the sequence of finite order linear processes (Xim)i by Theorem 3.3 we obtain Mn,m→dM as n→∞ in D↑d with the weak M1 topology.
If we show that for every ϵ>0limm→∞lim supn→∞P[dp(Mn,Mn,m)>ϵ]=0,
then by a generalization of Slutsky’s theorem (see Theorem 3.5 in [14]) it will follow that Mn→dM in D↑d with the weak M1 topology. Taking into account (6) and the fact that the metric dM1 on D↑1 is bounded above by the uniform metric, it suffices to show that
limm→∞lim supn→∞P(sup0≤t≤1|Mn(j)(t)−Mn,m(j)(t)|>ϵ)=0,
for every j=1,…,d, and further, as in the proof of Theorem 3.3, it is enough to show the last relation only for j=1. Denote by Jn,m the probability in the last relation (for j=1). Now we treat separately the cases α∈(0,1) and α∈[1,∞).
Case α∈(0,1). Recalling the definitions, we have
Jn,m≤P(⋁i=1n|Xi(1)−Xim(1)|an>ϵ)≤P(∑i=1n|Xi(1)−Xim(1)|an>ϵ).
Similarly as in the univariate case treated in [9] we obtain
Xi(1)−Xim(1)=∑j=1d(∑k=m+1∞Ck;1,jZi−k(j)+(Cm−1;1,j−C1,j(m,∨))Zi−m+1(j)+(Cm;1,j−C1,j(m,∧))Zi−m(j)),
and
∑i=1n|Xi(1)−Xim(1)|≤∑j=1d[∑i=−∞0|Zi−m(j)|∑s=1n‖Cm−i+s‖+(2∑l=m−1∞‖Cl‖)∑i=1n+1|Zi−m(j)|].
Therefore from (36) by applying condition (35) and the multivariate generalization of Lemma 3.2 in [8] (for the proof of this generalization see the arXiv preprint [6]) it follows that limm→∞lim supn→∞Jn,m=0, which means that Mn→dM as n→∞ in D↑d with the weak M1 topology.
Case α∈[1,∞). Define
Ak,j=Ck;1,j−C1,j(m,∨),ifk=m−1,Ck;1,j−C1,j(m,∧),ifk=m,Ck;1,j,ifk≥m+1,
for k≥m−1 and j∈{1,…,d}. Then using the representation of Xi(1)−Xim(1) obtained in the previous case we get
|Mn(1)(t)−Mn,m(1)(t)|≤⋁i=1n|Xi(1)−Xim(1)|an=⋁i=1n∑j=1d|∑k=m−1∞Ak,jZi−k(j)an|
for every t∈[0,1]. This, (35) and Lemma 5.2 in the arXiv preprint [6] yield limm→∞lim supn→∞Jn,m=0. Thus in this case also Mn→dM as n→∞ in D↑d with the weak M1 topology. □
When the sequence of coefficients (Ci) is deterministic, the limiting process M in Theorem 4.1 has the representation
M(t)=⋁Ti≤tPiSi,t∈[0,1],
where Si=(Si(1),…,Si(d)), with Si(k)=⋁j=1d(D+k,jQi(j)+∨D−k,jQi(j)−) for k=1,…,d. It is an extremal process with an exponent measure ρ, where for x∈[0,∞)d, x≠0,
ρ([[0,x]]c)=∫0∞P(y⋁k=1dS1(k)x(k)>1)αy−α−1dy.
A special case of multivariate linear processes studied in this paper is
Xi=∑j=0∞BjZi−j,i∈Z,
where (Bi)i≥0 is a sequence of random variables independent of (Zi). To obtain this linear process from the general one in (34) take
Ci;k,j=Bi,ifk=j,0,ifk≠j,
for i≥0 and k,j∈{1,…,d}. In this case, under the conditions from Theorem 4.1 the limiting process M reduces to
M(t)=(D˜+k,kM(k+)(t)∨D˜−k,kM(k−)(t))k=1,…,d=(B˜+M(k+)(t)∨B˜−M(k−)(t))k=1,…,d
for t∈[0,1], where (B˜+,B˜−) is a two-dimensional random vector, independent of (M(k+),M(k−))k=1,…,d∗, such that (B˜+,B˜−)=d(⋁i≥0Bi+,⋁i≥0Bi−). By an application of Propositions 5.2 and 5.3 in [14] we can represent M in the form
M(t)=B˜+M+(t)∨B˜−M−(t),t∈[0,1],
where M+(t)=(M(k+))k=1,…,d and M−(t)=(M(k−))k=1,…,d are extremal processes with exponent measures ν+ and ν− respectively, where for x∈[0,∞)d, x≠0,
ν+([[0,x]]c)=∫0∞P(y⋁k=1dQ1(k)+x(k)>1)αy−α−1dy
and
ν−([[0,x]]c)=∫0∞P(y⋁k=1dQ1(k)−x(k)>1)αy−α−1dy.
In the following example we show that the functional convergence in the weak M1 topology in Theorems 3.3 and 4.1 in general cannot be replaced by convergence in the stronger standard M1 topology.
Let (Ti)i∈Z be a sequence of i.i.d. unit Fréchet random variables, i.e. P(Ti≤x)=e−1/x for x>0. Take a sequence of positive real numbers (an) such that nP(T1>an)→1/2 as n→∞, for instance, we can take an=2n. Let
Zi=(T2i−1,T2i),i∈Z.
Then it follows easily that nP(‖Z1‖>an)→1 as n→∞. It is straightforward to see that the random process (Zi)i∈Z satisfies all conditions of Theorem 3.3, and hence the partial maxima processes Mn(·) of the linear process
Xi=C0Zi+C1Zi−1,i∈Z,
with
C0=1100andC1=0011,
converges in distribution in D↑2 with the weak M1 topology.
Next we show that Mn(·) do not converge in distribution under the standard M1 topology on D↑2. This shows that the weak M1 topology in Theorems 3.3 and 4.1 in general cannot be replaced by the standard M1 topology. Let
Vn(t)=Mn(1)(t)−Mn(2)(t),t∈[0,1],
where
Mn(1)(t)=⋁i=1⌊nt⌋Zi(1)+Zi(2)an=⋁i=1⌊nt⌋T2i−1+T2ian
and
Mn(2)(t)=⋁i=1⌊nt⌋Zi−1(1)+Zi−1(2)an=⋁i=1⌊nt⌋T2i−3+T2i−2an.
The first step is to show that Vn(·) does not converge in distribution in D1 endowed with the standard M1 topology. According to [15] (see also Proposition 2 in [1], where the term “weak M1 convergence” is used for convergence in distribution in the standard M1 topology) it suffices to show that
limδ→0lim supn→∞P(ωδ(Vn)>ϵ)>0
for some ϵ>0, where
ωδ(x)=supt1≤t≤t20≤t2−t1≤δM(x(t1),x(t),x(t2))
(x∈D1, δ>0) and
M(x1,x2,x3)=0,ifx2∈[x1,x3],min{|x2−x1|,|x3−x2|},otherwise.
Denote by i′=i′(n) the index at which max1≤i≤n−1Ti is obtained. Fix ϵ>0 and let An,ϵ={Ti′>ϵan} and
Bn,ϵ={Ti′>ϵanand∃k∈{−i′−1,…,3}∖{0}such thatTi′+k>ϵan/8}.
The regular variation property of T1 yields nP(T1>can)→(2c)−1 as n→∞ for c>0, and this, together with the fact that (Ti) is a sequence of i.i.d. variables, yield
limn→∞P(An,ϵ)=1−limn→∞(1−nP(T1>ϵan)n)n−1=1−e−(2ϵ)−1
and
lim supn→∞P(Bn,ϵ)≤lim supn→∞∑i=1n−1∑k=−nk≠03P(Ti>ϵan,Ti+k>ϵan/8)≤lim supn→∞(n−1)(n+3)P(T1>ϵan)P(T1>ϵan/8)=2ϵ−2.
Note that on the event An,ϵ∖Bn,ϵ it holds that Ti′>ϵan and Ti′+k≤ϵan/8 for every k∈{−i′−1,…,3}∖{0}. Now we distinguish two cases.
(i) i′ is an even number. Then i′=2i∗ for some integer i∗. Observe that on the set An,ϵ∖Bn,ϵ we have
Mn(1)(i∗n)=Ti′−1+Ti′an>ϵandMn(2)(i∗n)=⋁i=1i∗T2i−3+T2i−2an≤ϵ4,
and similarly
Mn(1)(i∗−1n)≤ϵ4andMn(2)(i∗−1n)≤ϵ4.
This implies
Vn(i∗n)=Mn(1)(i∗n)−Mn(2)(i∗n)>3ϵ4,
and
Vn(i∗−1n)=Mn(1)(i∗−1n)−Mn(2)(i∗−1n)∈[−ϵ4,ϵ4].
Further, on the set An,ϵ∖Bn,ϵ it holds that
Mn(1)(i∗+1n)=Ti′−1+Ti′anandMn(2)(i∗+1n)=Ti′−1+Ti′an,
which yields
Vn(i∗+1n)=0.
(ii) i′ is an odd number. Then i′=2i∗−1 for some integer i∗. Similarly as in the case (i) on the event An,ϵ∖Bn,ϵ one obtains
Vn(i∗n)>3ϵ4,Vn(i∗−1n)∈[−ϵ4,ϵ4]andVn(i∗+1n)=0.
Hence from (i) and (ii) we conclude that on the set An,ϵ∖Bn,ϵ it holds that
|Vn(i∗n)−Vn(i∗−1n)|>3ϵ4−ϵ4=ϵ2
and
|Vn(i∗+1n)−Vn(i∗n)|>3ϵ4.
Note that on the set An,ϵ∖Bn,ϵ one also has
Vn(i∗n)∉[Vn(i∗−1n),Vn(i∗+1n)],
and therefore taking into account (40) and (41) we obtain
ω2/n(Vn)≥M(Vn(i∗−1n),Vn(i∗n),Vn(i∗+1n))>ϵ2
on the event An,ϵ∖Bn,ϵ. Therefore, since ωδ(·) is nondecreasing in δ, it holds that
lim infn→∞P(An,ϵ∖Bn,ϵ)≤lim infn→∞P(ω2/n(Vn)>ϵ/2)≤limδ→0lim supn→∞P(ωδ(Vn)>ϵ/2).
Since x2(1−e−(2x)−1) tends to infinity as x→∞, we can find ϵ>0 such that ϵ2(1−e−(2ϵ)−1)>2, that is 1−e−(2ϵ)−1>2ϵ−2. For this ϵ, by relations (38) and (39), we have
limn→∞P(An,ϵ)>lim supn→∞P(Bn,ϵ),
i.e.
lim infn→∞P(An,ϵ∖Bn,ϵ)≥limn→∞P(An,ϵ)−lim supn→∞P(Bn,ϵ)>0.
This and (42) imply (37), and hence Vn(·) does not converge in distribution in D1 with the standard M1 topology.
To finish, if Mn(·) would converge in distribution in the standard M1 topology on D↑2, and then also on D2, using the fact that linear combinations of the coordinates are continuous in the same topology (see Theorems 12.7.1 and 12.7.2 in [17]) and the continuous mapping theorem, we would obtain that Vn(·)=Mn(1)(·)−Mn(2)(·) converges in D1 with the standard M1 topology, which is impossible, as is shown above.
Acknowledgments
The author would like to thank the referee for valuable comments and suggestions, which helped to improve the paper.
ReferencesAvram, F., Taqqu, M.: Weak convergence of sums of moving averages in the α–stable domain of attraction. Ann. Probab.20, 483–503 (1992) MR1143432Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl.119, 1055–1080 (2009) MR2508565. https://doi.org/10.1016/j.spa.2008.05.004Basrak, B., Tafro, A.: A complete convergence theorem for stationary regularly varying multivariate time series. Extremes19, 549–560 (2016) MR3535966. https://doi.org/10.1007/s10687-016-0253-5Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997) MR1464694Krizmanić, D.: Functional limit theorems for weakly dependent regularly varying time series. Ph.D. dissertation, University of Zagreb, Croatia, https://www.math.uniri.hr/~dkrizmanic/DKthesis.pdf. Accessed 28 June 2024.Krizmanić, D.: Skorokhod M1 convergence of maxima of multivariate linear processes with heavy-tailed innovations and random coefficients. arXiv preprint, https://arxiv.org/abs/2208.04054, 2022. Accessed 28 June 2024.Krizmanić, D.: Functional weak convergence of partial maxima processes. Extremes19, 7–23 (2016) MR3454028. https://doi.org/10.1007/s10687-015-0236-yKrizmanić, D.: Functional convergence for moving averages with heavy tails and random coefficients. ALEA Lat. Am. J. Probab. Math. Stat.16, 729–757 (2019) MR3949276. https://doi.org/10.30757/alea.v16-26Krizmanić, D.: Maxima of linear processes with heavy-tailed innovations and random coefficients. J. Time Ser. Anal.43, 238–262 (2022) MR4400293. https://doi.org/10.1111/jtsa.12610Kulik, R., Soulier, P.: Heavy-Tailed Time Series. Springer, New York (2020) MR4174389. https://doi.org/10.1007/978-1-0716-0737-4Lamperti, J.: On extreme order statistics. Ann. Math. Stat.35, 1726–1737 (1964) MR0170371. https://doi.org/10.1214/aoms/1177700395Mikosch, T., Wintenberger, O.: Extreme Value Theory for Time Series. Models with Power-Law Tails. Springer, New York (2024) MR4823721. https://doi.org/10.1007/978-3-031-59156-3Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987) MR0900810. https://doi.org/10.1007/978-0-387-75953-1Resnick, S.I.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007) MR2271424Skorohod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl.1, 261–290 (1956) MR0084897Tyran-Kamińska, M.: Convergence to Lévy stable processes under some weak dependence conditions. Stoch. Process. Appl.120, 1629–1650 (2010) MR2673968. https://doi.org/10.1016/j.spa.2010.05.010Whitt, W.: Stochastic-Process Limits. Springer, New York (2002) MR1876437